He left the game for good with less than six minutes remaining in the first half after he was tackled at Arizona's 44-yard line following a 4-yard run.
The sophomore already had several cuts on his throwing hand from past games then had trouble gripping the ball on Saturday.
Whether he'll be able to play against the Trojans hasn't been decided, Whittingham said.
Whittingham said he wanted to see how Wilson looked Tuesday in practice, then make a decision.
"It's a medical staff decision as well as Travis'," he said.
Schulz finished the game 12 of 23 for 142 yards and a touchdown. He would get the starting nod if Wilson can't play, but is sidelined only short term.
If Wilson is out longer, the Utes would consider taking freshman Conner Manning out of his redshirt year if he beats out Schulz in practice, Whittingham said.
Both will get increased snaps this week in practice.
Whittingham said a possible extended absence from the lineup for Wilson would be a "worst-case scenario."
The Utes have been through several of those scenarios in recent years, with current offensive coordinator Brian Johnson the Utah last quarterback to start and finish a year, doing so in 2008.
Since then, the Utes either had to find replacements for Jordan Wynn when he was injured or switched quarterbacks in hopes of reviving the offense after poor efforts.
Before the season started, Whittingham said it was crucial to have a healthy quarterback who could survive a season to help the offense be consistent. While he maintains he still thinks it is doable, the Utes seem almost snakebit.
If Schulz is the new quarterback, the Utes will rally around him just like they have others, defender Trevor Reilly said.
"I've been through it a lot the last couple years and if it happens, it happens," Reilly said. "Travis is talented and Schulz is talented, too. Both are D-I players and have the ability to do it."
Whittingham said if Schulz is the starter, Utah's offense will have to make some changes since Schulz has a good arm but isn't as mobile as Wilson.
"You have to take advantage of the guy who pulls the trigger," Whittingham said.
The Utes can take some comfort in knowing that USC is dealing with its share of injuries as well.
Tight ends Xavier Grimble and Randall Telfer as well as running back Justin Davis all sustained injuries in the Trojans' 14-10 loss to Notre Dame.
Star receiver Marqise Lee is listed as questionable as he continues to be hampered by a knee injury that forced him to the sidelines for the second half of Saturday's game.
The injuries leave USC with just one scholarship tight end in sophomore Jalen Cope-Fitzpatrick.
Utah at USC
P Saturday, 2 p.m.
TV • Pac-12 Networks